Simulating absorbance for spherical particle under plane wave illumination

[Vadim Zakomirnyi]

Hi all,

I am trying to reproduce absorbance for simple case of spherical nanoparticle on a substrate under plane wave illumination. According to the literature [see an example in attachments], absorbance is given by following equation: 

A = -log(I_total/I_ini), 

where I_total - is intensity of sum of scattered and initial field in aperture, and I_ini - is intensity of initial field in aperture. In other words, I need to get ratio of the intensity of the fields with and without a particle. The main problem is that I have to sum the field and only then calculate far field intensities in some aperture.

I`m trying to use SMUTHI for my project. So, I_total I have tried to sum smuthi.field.PlaneWaveExpansions objets of scattered and initial fields, and then find a far field of this sum with pwe_to_ff_conversion. For scattered field I used scattered_field_pwe for upgoing and downgoing fields separately. For initial field I used initial_field.plane_wave_expansion for top and bottom layer accordingly. Then I failed when tried to sum those objects because "ValueError: Plane wave expansion are inconsistent". ​Also, initial_field.plane_wave_expansion for my initial plane wave looks strange because of only single zero element for azimuthal_angles parameters array. Thus, I cant even get an angular dependence of far field intensity of the initial field (I_ini). I think, this is because I have a plane wave as initial field, but I`m not entirely sure.

Could someone please help me with my problem?  Is there a simple way to extract angularly dependent far field intensities of the total and initial fields?

Best,

Vadim Zakomirnyi

[Amos Egel]

Hi Vadim,

I understand that you would like to analyze the power radiated into a certain set of directions both for the total field and for the initial field.

But then you also mention the concept of an aperture. So I guess what you are actually interested in is the power radiated through some surface (maybe the entrance of some measurement equipment) at large but finite distance. Right?

I don't really understand why the thing you are trying to compute is called "absorbance". I think "extinction" is a more suitable term because it includes the absorbed power but also the power that is scattered into different directions than into the aperture.

If I got everything right, what I would recommend is to use the near field rather than the far field to compute the power flux. You can specify a set of positions in space (that represent the aperture surface) and then evaluate the electric and magnetic field vectors at these positions and finally compute the Poynting flux through the surface by numerical integration. I think that Dominik Theobald has done similar things in the past. Unfortunately, it might take longer simulation time because the near field is more heavy to compute than the far field. An NVIDIA gpu (can be a cheap one) can accelerate things a bit.

The thing with the far field intensity approach that you followed is that a plane wave is actually a quite complicated thing when it comes to far field intensity. In fact, a plane wave is an idealization of a beam that is infinitely wide with regard to its lateral extent and infinitely narrow when it comes to the spread in propagation direction. In other words, its plane wave spectrum is a delta-function. To see this, you can verify that if you insert a delta-function for the plane wave spectrum g in equation (2.18) of my dissertation, the result is a single plane wave.

As a consequence, the far field intensity in terms of "radiated power per solid angle" is not defined for a plane wave. The power radiated by a plane wave into a small solid angle interval is zero into almost all directions and infinite for that one direction that is along the k-vector of the plane wave.

This is why the concept of scattering or absorption cross sections need to be introduced when we talk about far field quantities in the context of plane wave excitation. They allow to relate the scattered field (which has a finite, continuous plane wave spectrum) to a plane wave (which has a delta-function plane wave spectrum).

An additional consequence is the following: If you want to model the power flux through an aperture in space for a field with continuous plane wave spectrum, you can assume that if the aperture is far away from the origin of radiation, the power radiated through the aperture is just the corresponding angular part of the far field intensity, i.e., you would integrate the far field intensity over all angles under which the aperture appears to the origin of radiation (for a proof of that assumption, one could use the stationary phase approximation that relates the electromagnetic field at some location far away from the source to the amplitude of the plane wave spectrum of the field at the direction that points to that location). However, if you have a single plane wave, this approach cannot be applied anymore: As the plane wave's lateral extent is infinite, the power passing some aperture (even if far away) is given by the aperture area times the power-per-area of the plane wave.

Does this make sense to you? Please don't hesitate to write back if something is unclear or if I have misunderstood your question.

Best regards, Amos

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Smuthi project repository: https://gitlab.com/AmosEgel/smuthi
Online documentation https://smuthi.readthedocs.io/en/latest/
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[Amos Egel]

P.S.: If the aperture is very far away from the particle, it might make sense to use the stationary phase approximation for the scattered field. It saves a lot of effort, and for the usual approach, a very fine sampling may be necessary.

[Krzysztof Czajkowski]

Hi Vadim,

do you really need far field angular spectrum to get absorbance? Can't you simply calculate extinction and scattering cross-sections, find their difference and normalize adequatly? There is also one thing to note when you think about absorbance of a single particle illuminated by a plane wave. Such absorbance may be larger than unity. For great explanation see https://aapt.scitation.org/doi/10.1119/1.13262

Feel free to ask more questions.

Best,

Krzysztof

[Vadim Zakomirnyi]

Dear all,

Thanks for a quick response! I appreciate you joining this discussion.

To Amos Egel:

Your answer definitely makes sense. Indeed, the radiation power of a plane wave exists only for the direction of the wave vector. However, I wasn't that interested in calculating this value. I needed the relationship between the far field with and without the particle system. I thought there would be a difference..

Anyway, please, let me include some details about my simulations. I think it will bring deeper understanding of my problem and may be will lead to elegant solution. It attachments you may found an article with great explanation of optical system I`m tried to model. As it shown in Fig.2(A), incoming wave front is nothing else then a collection of plane waves in some numerical aperture. To simulate this I simply create a cycle of smuthi.Simulations for every element of array of plane waves. Then, I calculate near field with graphical_output.compute_near_field and sum all the results. It takes time, but it works pretty well - I successfully reproduce an electric field distribution around particle. But I feel that this strategy does not work correctly for calculating absorbance spectra from Fig.2(C) from the same article. As you said, it make no sense to calculate a far field intensity in aperture for a plane wave source. But technically, I'm modeling some sector of the incoming spherical wave instead of a plane wave source. Therefore, don't you think it will be possible to extract solid angle dependence of the initial field, and as a consequence absorbance coefficients in the form of ration between far fields of the system with and without particle?

Also, what do you think about possibility to modify an initial plane wave source in a way when it will contain finite size array of plane waves for some aperture. With this approach I will have to run my simulations only once.  This will increase the speed of my simulations hundreds of times.  

 

I hope I explained my idea correctly. Please let me know if you missed anything.

To Krzysztof Czajkowski:

Thanks for sharing article. It I`m totally fine with absorbance greater than unity. Actually, those spectral features are the subject of my study too. And yes, I also though about extracting absorbance coefficient from extinction cross section and scattering cross section. Unfortunately, those values are not corresponds to the figures from article I attached. I still think that it is a different absorption, if it make sense. 

Best,

Vadim Zakomirnyi

[Amos Egel]

Yes, I would also recommend to use an initial field that already represents the beam you want to model rather than doing an individual simulation for every plane wave component.

Figure 2A shows a schematic of a beam, but it doesn't specify how the beam actually looks like. It is drawn as a wavefront that sharply ends at the sides over the full propagation length. You can create a beam that sharply ends outside some aperture for a given z-plane. You would start from the field pattern in that plane (I understand that it should be constant inside some aperture and zero outside) and then do a Fourier transform to calculate the plane wave spectrum. But be aware that this beam would fulfill that condition only at the given z-plane (its "focus"). At other z-positions it would diverge and no longer be zero outside the aperture ( -> diffraction).

I am not even convinced that this kind of beam is implied by the drawing in figure 2A. Maybe the authors just wanted to draw some beam and didn't care to draw a realistic beam shape.

Have you considered using a Gaussian beam? They are a widely used approximation to laser beams and are already implemented in Smuthi.

Alternatively, if you stick to the idea of defining a wavefront that is constant in your aperture and zero outside, it will be straightforward to implement that into smuthi. You would need to write a new initial field class that should derive from smuthi.initial_field.InitialPropagatingWave. All you need to do is to override the plane_wave_expansion method with an appropriate expansion. To this end, you would need to do some math and find out an analytical expression for the function g(kappa, alpha) of coefficients for the plane wave expansion of the beam (it will be related to but not identical with the Fourier transform of the wavefront that you have in mind). I can give you further assistance, if you decide to follow that route.

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